• The Virial Theorem postulates that for self-gravitating systems:
  • for an N-particle system;
  • for gaseous objects

  where n_vir is a positive number, T is the kinetic, U_int the internal energy and W the total gravitational (potential) energy and <..> denotes the time-average. In other words, the virial theorem holds when the system has relaxed into a steady configuration.

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  Proof goes along these lines. The quantity

  time-averages to 0. Q (the "virial") is itself the derivative of the moment of inertia. For a gravitating gas, we derive from the hydrostatic balance equation

  To proceed, must be linked to the internal energy per unit mass . For an ideal gas this is just

  where is the number of free degrees and is the heat capacity ratio. For a mono-atomic gas, , and .

• The Virial Theorem postulates that for self-gravitating systems:
  • for an N-particle system;
  • for gaseous objects

  where n_vir is a positive number, T is the kinetic, U_int the internal energy and W the total gravitational (potential) energy and <..> denotes the time-average. In other words, the virial theorem holds when the system has relaxed into a steady configuration.

• Specific examples include:
  •   (fully ionized or mono-atomic gas)
  •   (radiation gas)

• The Virial Theorem postulates that
  • for an N-particle system;
  •   (fully ionized or mono-atomic gas)
  •   (radiation gas)
• The virial theorem is not the same as energy conservation!

  Conservation of energy dictates that where the total energy is conserved in an isolated system. Hence we obtain that objects in virial equilibrium obey (a different prefactor may apply if the gas is not mono-atomic).
  Note — the expression for the virial theorem can be extended to include magnetic support, rotational support etc.

• The Virial Theorem postulates that
  •   (fully ionized or mono-atomic gas)
  •   (radiation gas)
• The virial theorem is not the same as energy conservation!
• Applications:
  • (Proto-)stars heat up when they contract:

    Protostars (no nuclear burning) lose energy. Of the gravitational energy liberated during contraction, half is radiated and the other half goes into internal energy. Stars heat up while loosing energy!
    This corresponds to moving from state A → B in the figure right

  • Stars are thermodynamically stable

    They expand and cool down when their total energy E increases (!). This corresponds to moving from B → A in the figure right. This kind of "virial stability" only applies to the ideal gas law.

  • Obtain the (dynamical) mass by measuring the linewidth.

The condition for a spherical, uniform cloud to collapse is when its mass exceeds the Jeans mass

When the total energy of the cloud decreases when it is perturbed. If we insert numerical constants for the numbers and write the densities in terms of the number density we obtain

Note some (giant) clouds have suggesting that the should collapse! In reality, these clouds feature additional support mechanisms (like magnetic support). Hence the condition is a necessary but not (always) sufficient condition for collapse.

If the cloud is out of equilibrium, it collapses on a free-fall time

• During collapse of the cloud, ρ increases and the Jeans mass decreases. The cloud fragments with the fragments collapsing on their own.
• The isothermal collapse, requires efficient cooling.
• An increase in temperature, on the other hand, opposes cloud fragmentation, causing M_Jeans to rise

• An increase in temperature, on the other hand, opposes cloud fragmentation, causing M_Jeans to rise

By balancing the heat generated by the collapse with the expression for blackbody radiation applicable when the cloud starts to cool less efficienty ( becomes adiabatic) we can solve for the mass where the fragmentation terminates

where I have omitted numerical factors and ε~0.1 is an efficiency factor. For T=10³ K, we obtain .

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This is of course an extremely crude estimate. But it tells that the protostar that forms at the end of the collapse state are of stellar mass — not of planet mass or the mass of galaxy. Further accretion onto this protostar will of course occur.

Initial mass function (IMF)

— read CO p.430 —

The Initial Mass Function or simply IMF is the distribution of stars formed by stellar mass

So ξ(m)Δm gives the number of stars that are formed in mass interval [m,m+Δm]

The IMF follows (steep) power-laws at high masses, but it turns over at low stellar masses (it has to!)

The origin of the IMF is an area of active research in star formation

1. Dark cores form and become unstable when their gravitational potential energy dominates thermal energy:
   (mass exceeds Jeans mass)
2. The collapse takes place on the free-fall timescale
   this is the embedded phase Class 0

3. A star has formed. Class 1. An outflow develops, while material is still accreting onto the star. A disks forms.
4. Accretion and outflow weaken. This is the stage where planets are thought to form. Class 2 or the T-Tauri phase.
5. After the gas clears, a debris disks may be left

   Debris disk consist of solid material that did not assemble into planetary bodies. In the solar system the asteroid belt and the Kuiper belt are debris disks

6. A planetary system emerges. The final architecture of the planet system is set when orbital stability is reached.

The fluid equations:

where the E.o.S is chosen as isothermal (constant temperature), such that c_s — the isothermal sound speeed — is constant.

To 0^th order, for a uniform cloud, the solutions to these equations read simply:

A linearly stability analysis quantifies how the system reacts to small perturbations that are wave-like in nature, e.g.,

with k the wavenumber and ω the frequency (or growth rate when it is imaginary!). Without loss of generality we have assumed that the wave travels in the x-direction. Note than ω can be complex, so the wave can grow (or decay)!

After some algebra, you arrive at the perturbation equations

The solution to the linearly stability analysis is the dispersion relationship:

A dispersion relationship is a relation between the scale of the perturbation (k) and the growth rate (ω).

We obtain:

• small scales (large k): ω² is positive: perturbations remain wave-like and do not grow!
• large scales (small k): ω² is negative: perturbations grow exponentially and are unstable!
• a critical scale

  You can verify that amounts to the Jeans mass, barring a numerical coefficient

For disks the steady solution reads:

the assumption here is that the disk is thin. It does not need to be Keplerian rotating, but it should satisfy Also, we consider that the disk is axisymmetric (quantities do not depend on φ) and consider radial perturbations only (k oriented in the radial direction).

... and the dispersion relationship is

... and the dispersion relationship is

We obtain the following:

• The most critical wavelength is 
• The disk becomes gravitationally unstable when the Toomre-Q parameter
• The corresponding mass is .

  Question:
  Is Q_T <1 the only criterion for gravitational instability to result in collapse?

... and the dispersion relationship is

We obtain the following:

• The most critical wavelength is 
• The disk becomes gravitationally unstable when the Toomre-Q parameter
• The corresponding mass is .
• In addition, gravitational collapse requires fast enough cooling

  Quantitatively . Otherwise the formed clumps will just be sheared apart

... and the dispersion relationship is

We obtain the following:

• The most critical wavelength is 
• The disk becomes gravitationally unstable when the Toomre-Q parameter Q_T<1.
• In addition, gravitational collapse requires fast enough cooling

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• For dust particles, can be used, amounting to a critical scale of

Hence, particles need to settle. The settling of particles is promoted by the stellar gravity, but opposed by turbulence.

Smooth Particle Hydrodynamics (SPH) simulations of gravitational instability in disks

1. standard (fast cooling)
2. slow cooling

Smooth Particle Hydrodynamics (SPH) simulations of gravitational instability in disks

1. standard (fast cooling)
2. slow cooling

Properties

• disks are naturally formed as byproduct of star formation due to angular momentum conservation
• disks are usually dusty, obscuring the interior disk midplane regions

Properties

• disks are naturally formed as byproduct of star formation due to angular momentum conservation
• disks are usually dusty, obscuring the interior disk midplane regions
• the dust results in an IR-excess

  the total integrated emission is a superposition of several black body curves. The IR-excess is how disks were historically detected

Properties

• disks are naturally formed as byproduct of star formation due to angular momentum conservation
• disks are usually dusty, obscuring the interior disk midplane regions
• the dust results in an IR-excess

Properties

• disks are naturally formed as byproduct of star formation due to angular momentum conservation
• disks are usually dusty, obscuring the interior disk midplane regions
• the dust results in an IR-excess
• Minimum-mass solar nebula (MMSN)

  By spreading out the heavy element mass of the planets over rings, Weidenschilling (and Hayashi) obtained a crude prescription of the dust and gas density during the time of the formation of the planets. This prescription — known as the Minimum-Mass Solar Nebula — has many flaws, but serves as a useful benchmark. A frequently-used profile is:

→ Reconstruction of the early disk surface density from the current position of the planets in the solar system. Weidenschilling (1977)

Properties

• disks are naturally formed as byproduct of star formation due to angular momentum conservation
• disks are usually dusty, obscuring the interior disk midplane regions
• the dust results in an IR-excess
• Minimum-mass solar nebula (MMSN)
• disks are observed to accrete onto their host star at rates of  

  Mass and angular momentum are being transported.

Properties

• disks are naturally formed as byproduct of star formation due to angular momentum conservation
• disks are usually dusty, obscuring the interior disk midplane regions
• the dust results in an IR-excess
• Minimum-mass solar nebula (MMSN)
• disks are observed to accrete onto their host star at rates of  

  Mass and angular momentum are being transported.

Properties

• disks are naturally formed as byproduct of star formation due to angular momentum conservation
• disks are usually dusty, obscuring the interior disk midplane regions
• the dust results in an IR-excess
• Minimum-mass solar nebula (MMSN)
• disks are observed to accrete onto their host star at rates of  
• Accretion terminates over timescales of ~Myr

  A natural explanation is that the gas disk is gone after several Myr. It sets a timescale for the formation of gas giant planets

There are two main paradigms for the formation of planets:

1. disk instability: planets form following a gravitational instability in the gas disk
2. core accretion: planet formation occurs bottom-up by coagulation (collisions between solid particles)

   —initially, solids stick due to surface forces
   —later, bodies are held together by gravity
   (at intermediate sizes sticking is problematic)